Vectors Questions with Solutions

A bird is flying in a straight line with velocity vector $10\hat{i}+6\hat{j}+\hat{k}$, measured in km/hr. If the starting point is $\left(1,2,3\right)$, how much time does it take to reach a point in space that is 13 m high from the ground?

The values of the non-zero scalars $\alpha$ and $\beta$ such that, for all vectors $\hat{a}$ and $\hat{b}$, $\alpha \left(2\hat{a}-\hat{b}\right)+\beta \left(\hat{a}+2\hat{b}\right)=8\hat{b}-\hat{a}$, are

A force of 78 grams (weight) acts at the point (2, 3, 5), the direction ratios of its line of action being 2, 2, 1. The magnitude of its moment about the line joining the origin to the point (12, 3, 4) is

The position vectors of the vertices A, B, C of a tetrahedron ABCD are $\hat{i}+\hat{j}+\hat{k}$, $\hat{i}$ and $3\hat{i}$ respectively, and the altitude from the vertex D to the opposite face ABC meets the face at E. If the volume of the tetrahedron is $\frac{2\sqrt{2}}{3}$, then the length of DE is:

$\hat{a}$ and $\hat{b}$ are non-zero, non-collinear vectors such that $\left|\hat{a}\right|=2$, $\hat{a}\cdot \hat{b}=1$ and the angle between $\hat{a}$ and $\hat{b}$ is $\frac{\pi }{3}$. If $\hat{r}$ is any vector satisfying $\hat{r}\cdot \hat{a}=2$, $\hat{r}\cdot \hat{b}=8$, $\left(\hat{r}+2\hat{a}-10\hat{b}\right)\cdot \left(\hat{a}\times \hat{b}\right)=6$ and $\hat{r}+2\hat{a}-10\hat{b}=\lambda \left(\hat{a}\times \hat{b}\right)$, then $\lambda =$

The value of $m$ for which the volume of the parallelepiped is 4 cubic units, whose three edges are represented by $\hat{a}=m\hat{i}+\hat{j}+\hat{k}$, $\hat{b}=\hat{i}-\hat{j}+\hat{k}$ and $\hat{c}=\hat{i}+2\hat{j}-\hat{k}$, is

The number of distinct real values of $\lambda$ for which the vectors ${\lambda }^2\hat{i}+\hat{j}+\hat{k}$, $\hat{i}+{\lambda }^2\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+{\lambda }^2\hat{k}$ are coplanar is

If $\left|\hat{F}\right|=40$ N, $\left|\hat{D}\right|=3$ m and $\theta ={60}^{\circ }$ is the angle between them, then the work done by $\hat{F}$ acting through the displacement $\hat{D}$ is

A man starts at the origin O and walks a distance of 3 units in the north-east direction and then walks a distance of 4 units in the north-west direction to reach the point P. Then $\widehat{OP}$ is equal to

How much work does it take to slide a crate for a distance of 25 m along a loading dock by pulling on it with a 180 N force directed at an angle of ${45}^{\circ }$ to the dock?

The vector $\hat{A}=\left(2x+1\right)\hat{i}+\left(x^2-6y\right)\hat{j}+\left(x{y}^2+3z\right)\hat{k}$ is a

The length of the projection of $\hat{a}=2\hat{i}+3\hat{j}+\hat{k}$ on $\hat{b}=-2\hat{i}+\hat{j}+2\hat{k}$ is equal to:

If $\hat{a}$ and $\hat{b}$ are two vectors such that $\left|\hat{a}\right|=3$, $\left|\hat{b}\right|=4$ and $\left|\hat{a}+\hat{b}\right|=1$, then the value of $\left|\hat{a}-\hat{b}\right|$ is

If $\hat{a}=\hat{i}+\hat{j}+\hat{k}$, $\hat{b}=2\hat{i}-\hat{j}+3\hat{k}$ and $\hat{c}=\hat{i}-2\hat{j}+\hat{k}$, then a vector of magnitude $\sqrt{22}$ which is parallel to $2\hat{a}-\hat{b}+3\hat{c}$ is

Let $\hat{a}=2\hat{i}-3\hat{j}+4\hat{k}$, $\hat{b}=\hat{i}+2\hat{j}-\hat{k}$ and $\hat{c}=3\hat{i}+\hat{j}+\lambda \hat{k}$ be the co-terminal edges of a parallelepiped whose volume is 5 units. Then the value of $\lambda$ is